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lecture02

# lecture02 - 1 Distances(again t(s 0 v 30 Give two different...

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1 Distances (again) Example 1.1. Speedometer readings for a motorcycle at 12 second intervals are given: t ( s ) 0 12 24 36 48 60 v 30 28 25 22 24 27 Give two diﬀerent estimates. 2 Deﬁnite Integral Recall what we just did. Deﬁnition 1. If f is a continuous function deﬁned for a x b , we divide the interval [ a, b ] into n subintervals of equal width Δ x = ( b - a ) /n . We let x 0 = 1 , x 1 , . . . , x n = b be the endpoints of the intervals and we let x * 1 , x * 2 , . . . , x * n be any sample points in these intervals, so x * i is in [ x i - 1 , x i ]. The deﬁnite integral of f from a to b is Z b a f ( x ) dx = lim n →∞ n X i =1 f ( x * i x To make things easier, we usually pick the same point in every interval, usually either the left endpoint or the right endpoint. So usually x * i = x i or x * i = x i - 1 . Also, as long as f is continuous, it doesn’t matter which one we pick. They’ll always end up the same. Deﬁnition 2. In R b a f ( x ) dx , the f ( x ) is called the integrand. The a and b are limits of integration - a is the lower limit and b is the upper limit. dx has no oﬃcial meaning by itself. It’s pretty much just a symbol.

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lecture02 - 1 Distances(again t(s 0 v 30 Give two different...

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